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Math Help - Proof for some exponent rules

  1. #1
    Senior Member Spec's Avatar
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    Proof for some exponent rules

    I'm looking for proof for the following exponent rules:

    x^px^q = x^{p+q}
    x^p/x^q = x^{p-q}
    x^py^p = (xy)^p

    Without using any logarithmic functions and not through induction.
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  2. #2
    Eater of Worlds
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    Here's a way to look at it. I hope it's allowed.

    x^{p}x^{q}=\underbrace{x\cdot{x}\cdot{x}\cdot{x}\c  dot{x}....\cdot{x}}_{\text{p factors of x}}\underbrace{\cdot{x}\cdot{x}\cdot{x}\cdot{x}...  ...\cdot{x}}_{\text{q factors of x}}

    Since the total number of factors of x on the right is p+q, this is equal to

    x^{p+q}


    Will that suffice?.
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  3. #3
    Senior Member Spec's Avatar
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    Thanks for the reply!

    I'm not quite sure that would qualify as a proof. Wouldn't it count more as induction?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Spec View Post
    Thanks for the reply!

    I'm not quite sure that would qualify as a proof. Wouldn't it count more as induction?
    i think its valid. wouldn't really describe it as an induction
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  5. #5
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    The proof is okay. You can do it with induction if you really want, but there is no point. It is simple enough itself.
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  6. #6
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    hmmmmmmm
    Maybe you should throw in some values (positive and negative) as that would certainly classify as proof.

    1a) Let x = 1, p = 2, q = 3
    1*1 = 1, 15 = 1 therefore
    Let x = -2, p = 2, q = 3
    -2*-2 = -32, -25 = -32 therefore

    1b) Let x = 1, p = 2, q = 3
    1/1 = 1, 1-1 = 1 therefore
    Let x = -2, p = 2, q = 3
    -2/-2 = -0.5, -2-1 = -0.5 therefore

    1c) Let x = 1, y = 2, p = 3
    1*2 = 8, (1*2) = 8 therefore
    Let x = -2, y = 3, p = 3
    -2*3 = -216, (-2*3) = -216 therefore

    If you prefer you can use your own values as long as one is negative and one is positive. This would definitely class as proof as far as i can see.
    Hope this helps.
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Sean12345 View Post
    hmmmmmmm
    Maybe you should throw in some values (positive and negative) as that would certainly classify as proof.

    1a) Let x = 1, p = 2, q = 3
    1*1 = 1, 15 = 1 therefore
    Let x = -2, p = 2, q = 3
    -2*-2 = -32, -25 = -32 therefore

    1b) Let x = 1, p = 2, q = 3
    1/1 = 1, 1-1 = 1 therefore
    Let x = -2, p = 2, q = 3
    -2/-2 = -0.5, -2-1 = -0.5 therefore

    1c) Let x = 1, y = 2, p = 3
    1*2 = 8, (1*2) = 8 therefore
    Let x = -2, y = 3, p = 3
    -2*3 = -216, (-2*3) = -216 therefore

    If you prefer you can use your own values as long as one is negative and one is positive. This would definitely class as proof as far as i can see.
    Hope this helps.
    i'm afraid citing specific examples does not constitute proof, at least not as far as mathematics is concerned
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  8. #8
    Eater of Worlds
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    The proof I gave is satisfactory.
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  9. #9
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    I always thought that was an induction Galactus.
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